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Tilt Aftereffect and Adaptation-Induced Changes in Orientation Tuning inVisual Cortex

Dezhe Z. Jin,1 Valentin Dragoi,2 Mriganka Sur,3 and H. Sebastian Seung4

1Department of Physics, Pennsylvania State University, University Park, Pennsylvania; 2Department of Neurobiology and Anatomy,University of Texas Medical School at Houston, Houston, Texas; 3Picower Center for Learning and Memory and 4Howard HughesMedical Institute and Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts

Submitted 1 June 2004; accepted in final form 27 August 2005

Jin, Dezhe Z., Valentin Dragoi, Mriganka Sur, and H. SebastianSeung. Tilt aftereffect and adaptation-induced changes in orientationtuning in visual cortex. J Neurophysiol 94: 4038–4050, 2005. Firstpublished August 31, 2005; doi:10.1152/jn.00571.2004. The tilt af-tereffect (TAE) is a visual illusion in which prolonged adaptation toan oriented stimulus causes shifts in subsequent perceived orienta-tions. Historically, neural models of the TAE have explained it as theoutcome of response suppression of neurons tuned to the adaptingorientation. Recent physiological studies of neurons in primary visualcortex (V1) have confirmed that such response suppression exists.However, it was also found that the preferred orientations of neuronsshift away from the adapting orientation. Here we show that addingthis second factor to a population coding model of V1 improves thecorrespondence between neurophysiological data and TAE measure-ments. According to our model, the shifts in preferred orientation havethe opposite effect as response suppression, reducing the magnitude ofthe TAE.

I N T R O D U C T I O N

The tilt aftereffect (TAE) is a striking visual illusion inwhich prolonged adaptation to an oriented visual stimuluscauses subsequent stimuli to appear rotated away from theadapting orientation (Gibson and Radner 1937; He and Mac-Leod 2001; Magnussen and Johnsen 1986; Mitchell and Muir1976). Explaining this and other aftereffects in terms of neuralmechanisms has been an important outstanding problem. His-torically, a popular explanation of the TAE has been a hypoth-esized relative suppression of neurons tuned to the adaptingorientation (Clifford et al. 2000; Coltheart 1971; Sutherland1961; Wainwright 1999). Recent physiological studies haveconfirmed that adaptation leads to suppression of neural re-sponses near the adapting orientation. These experiments havealso identified an additional effect of adaptation: the preferredorientations of neurons repulsively shift away from the adapt-ing orientation (Dragoi et al. 2000, 2001). Here we construct apopulation coding model that includes both factors and showthat the repulsive shifts of preferred orientations are importantfor quantitatively explaining the TAE. According to the model,the TAE is indeed caused by the relative suppression of neuralresponses. However, it is substantially weakened by the pre-ferred orientation shifts. We suggest that the visual system usesthe repulsive shifts of preferred orientations to reduce theperceptual error in orientation that could be induced by neuralresponse suppression.

Quantitative measurements of the TAE are schematicallysummarized in the graph of Fig. 1A, which depicts the differ-ence between the perceived and true orientation of a teststimulus as a function of the difference between the test andadapting orientations. According to this graph, the perceivedorientation is similar to the true orientation, but rotated awayfrom the adapting orientation by up to 4 deg (Clifford et al.2000; Gibson and Radner 1937; Magnussen and Johnsen 1986;Mitchell and Muir 1976). That is, the adapting orientation“repels” the perceived orientation.

Neurons in the primary visual cortex (V1) respond selec-tively to the orientation of a stimulus (Hubel and Wiesel 1962).Orientation selectivity is generally characterized by a tuningcurve depicting the firing rate of a neuron as a function ofstimulus angle. Two major changes in tuning curves of VIneurons are observed after adaptation, particularly when theadapting orientation is close to a cell’s preferred orientation(Dragoi et al. 2000) (Fig. 1B). First, the amplitude of the tuningcurve on the flank near the adapting orientation depresses afteradaptation; this is often accompanied by an increase in re-sponse amplitude on the opposite or far flank. Second, thelocation of the peak response, or preferred orientation of thecell, shifts away from the adapting orientation, so that theeffect is “repulsive.” The amount of the shift, which dependson the difference between the preferred and the adaptingorientations, can be on the order of 10 deg (Dragoi et al. 2000)(Fig. 1B).

To show that the adaptation-induced changes of the tuningcurves of V1 neurons are quantitatively consistent with theTAE, we mathematically analyze a population coding model.Similar to previous models (Clifford et al. 2000; Gilbert andWiesel 1990; Pouget et al. 2000; Vogels 1990; Wainwright1999), our model assumes that the population response profileof V1 neurons to a stimulus determines its perceived orienta-tion. The analysis unveils a quantitative relationship betweenadaptation-induced changes of the perceived orientations andthose of the tuning curves: The amplitude suppression of thetuning curves near the adapting orientation is positively corre-lated with the sum of the repulsive shifts of perceived orien-tations and the preferred orientations of neurons. We use thisquantitative relationship to check the consistency between thepsychophysical and physiological data. From the measuredamount of the TAE and the shift of the preferred orientations,we predict the response suppression of the tuning curves near

Address for reprint requests and other correspondence: D. Jin, Departmentof Physics, The Pennsylvania State University, 104 Davey Lab, PMB #206,University Park, PA 16802 (E-mail: [email protected]).

The costs of publication of this article were defrayed in part by the paymentof page charges. The article must therefore be hereby marked “advertisement”in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

J Neurophysiol 94: 4038–4050, 2005.First published August 31, 2005; doi:10.1152/jn.00571.2004.

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the adapting orientation. This prediction is then compared withthe observed response suppression. The results confirm that theTAE is quantitatively consistent with the measured changes oftuning curves of V1 neurons under adaptation. The relationshipfurther illustrates that, for a given response suppression, therepulsive shift of the preferred orientation reduces the amountof the TAE and thus the perception error.

The relationship between the changes of the orientationtuning curves and those of the orientation perceptions has beendiscussed qualitatively before (Clifford et al. 2000; Coltheart1971; Gilbert and Wiesel 1990; Sutherland 1961; Teich andQian 2003; Wainwright 1999; Yao and Dan 2001). However,such qualitative discussions cannot resolve the issue of theconsistency between the adaptation-induced changes of thetuning curves and the TAE; a quantitative analysis is neces-sary. Our novel mathematical analysis enables the quantitativecomparison between the physiological and psychophysicaldata.

M E T H O D S

Experimental data

In this paper we use data from physiological and psychophysicalexperiments documented previously. Here we briefly describe theseexperiments. In the physiological experiments (Dragoi et al. 2000),the orientation tuning curves of neurons in V1 of anesthetized catswere measured by presenting high-contrast square-wave drifting grat-ings at 16 orientations (equally spaced at 22.5 deg), and recording thespike trains. The gratings had a spatial frequency of 0.5 cycle/deg, andtemporal frequency of 1 Hz. Before adaptation, each orientation waspresented for ten trials, with each trial lasting 2.5 s. Neuron spike rateswere averaged over trials for each orientation. Adaptation was in-duced by presenting a drifting grating at the adapting orientation for2 min. After adaptation, each of the 16 orientations was presented for112 trials, with each trial lasting 2.5 s, preceded by a 5-s presentationof the adapting orientation to maintain the effects of adaptation.

In the psychophysical experiments that measured the TAE (Cliffordet al. 2000, 2003) human subjects were presented with contrastreversing, stationary sinusoidal gratings with spatial frequency 1cycle/deg and temporal frequency 1 Hz. The TAE was measured usingadapting stimuli at six orientations of equal spacing of 15 deg. Eachadapting stimulus was presented for 1 min. After adaptation, the

perceived vertical orientation was measured using test stimuli withorientations progressively closer to the subjects’ judgment of verticalorientation. Each testing stimulus lasted 400 ms followed by a 5-spresentation of the adapting stimulus to maintain the effects ofadaptation. The test stimuli were presented for 60 trials. The differ-ence between the true and perceived vertical orientation was the TAE.

The model

The central feature of our model is the rate function, which is acompact description of the tuning curves of the neuronal population.Our goal is to show how the activity of the neuron population, asdefined by the rate function, changes as a result of the responsesuppression and orientation shift of tuning curves. We describe threedifferent procedures for calculating, through the rate function, therelationship between the changes of the tuning curves and those ofpopulation responses, and illustrate our results in detail with onemethod, the winner-take-all method (see APPENDIX). We then showthat the other two methods, the population vector method and themaximum-likelihood method, lead to similar results.

R E S U L T S

Orientation perception is commonly presumed to arise fromthe population responses of V1 neurons to oriented stimuli(Clifford et al. 2000; Gilbert and Wiesel 1990; Pouget et al.2000; Vogels 1990; Wainwright 1999). From this perspective,it is straightforward to see how adaptation-induced changes inthe tuning curves of V1 neurons lead to errors in orientationperception because such changes alter the population responseprofiles of V1 neurons. Thus, the neural basis of the TAE issimple to grasp qualitatively. However, it is not obvious thatthe specific changes of tuning curves observed in physiologicalexperiments (Dragoi et al. 2000) are quantitatively compatiblewith the TAE. The experiments show that the preferred orien-tations of neurons shift away from the adapting orientation;moreover, maximum firing rates are reduced especially forneurons with preferred orientations near the adapting orienta-tion. Do these changes lead to shifts of the perceived orienta-tion away from the adapting orientation, as in the TAE?Moreover, are the amounts of change of tuning curves consis-tent with the magnitude of perception shifts in the TAE? We

FIG. 1. Summary of psychophysical and physiological datarelated to the effects of orientation adaptation. A: prolongedadaptation to a tilted grating (left) causes the vertical grating(right) to appear rotated away from the orientation of theadaptor (Gibson and Radner 1937; He and MacLeod 2001;Magnussen and Johnsen 1986; Mitchell and Muir 1976). Right:curve schematically summarizes the difference between the trueand adapting orientations. B: prolonged adaptation to a gratingcauses repulsive shifts of the preferred orientations of neuronsin primary visual cortex (V1) (Dragoi et al. 2000). Left: tuningcurves of 2 neurons. For each neuron, the gray curve representsthe tuning curve of a neuron before adaptation. Black curve isthe tuning curve after adaptation. Adapting orientation is rep-resented by the gray vertical line. Preferred orientation shiftsrepulsively away from the adapting orientation (horizontal ar-rows). Amplitude of the tuning curve also changes. For theneuron with preferred orientation close to the adapting orienta-tion, the amplitude diminishes (left vertical arrow). For theneuron with preferred orientation far from the adapting orien-tation, the amplitude may increase (right vertical arrow). Right:curve schematically summarizes the difference between thepreferred orientation after adaptation and the preferred orienta-tion before adaptation as a function of distance between thepreferred and adapting orientations.

4039AFTEREFFECT CHANGES IN ORIENTATION TUNING

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address these questions by mathematically analyzing the rela-tionship between the changes of tuning curves and those oforientation perception in a population coding model.

To do this, two issues about population coding models mustbe addressed. The first issue concerns the diversity of orienta-tion tuning properties of V1 neurons. Tuning curves of V1neurons, even for those with the same preferred orientations,may have quite different widths and maximum rates (Hubeland Wiesel 1962). Such diversity makes our mathematicalanalysis difficult. We overcome this difficulty by replacing allneurons of the same preferred orientation with a single “rep-resentative neuron.” The tuning curve of this single neuron isthe average of those of the neurons with the same preferredorientation. Thus, in our population coding model, there is oneneuron for each preferred orientation. Before adaptation, thetuning curves of all neurons have the same Gaussian shape.Each neuron is labeled with its preferred orientation in theunadapted state. After adaptation, the tuning curves remainGaussian; however, the preferred orientations may shift fromthe neuron labels. Moreover, the amplitudes and the widths ofthe tuning curves may change as well.

The second issue concerns how the rest of the brain “readsout” orientation from V1 neuron responses. This is not a settledmatter in population coding models in general (Pouget et al.2000). Among many possible proposals, three methods arecommonly used in the literature: the winner-take-all, the pop-ulation vector, and the maximum-likelihood methods. In thewinner-take-all method, the perceived orientation is set to thelabel of the neuron that fires maximally to the stimulus. In the

population vector method (Georgopoulos et al. 1982; Gilbertand Wiesel 1990; Vogels 1990), each neuron contributes atwo-dimensional vector with orientation equal to twice its labeland length equal to its firing rate; summation of these vectorsresults in a population vector, whose orientation is taken astwice the perceived orientation. In the maximum-likelihoodmethod (Paradiso 1988; Pouget et al. 2000), each perceivedorientation is associated with a predetermined template ofpopulation responses. These templates are compared to thepopulation response to a stimulus and the one that best matchesdetermines the perceived orientation. In our analysis of theneural basis of the TAE, we use all three methods and showthat they lead to similar results. The winner-take-all method isthe simplest and is amenable to mathematical analysis; wetherefore explain our results mostly in terms of this readoutmethod. The results from the other two methods are presentedlater and compared.

Before presenting the detailed quantitative analysis, weexplain qualitatively how adaptation-induced changes oftuning curves are related to the TAE. We first examine thetwo types of changes—response suppression and orientationshift—separately, and then combine them. Without loss ofgenerality, we assume that the adapting orientation is at�1 � 0.

Figure 2A shows that the suppression of tuning-curve am-plitudes near the adapting orientation causes repulsive shifts ofthe perceived orientations, which is the essence of the fatiguemodel of TAE (Clifford et al. 2000; Coltheart 1971; Sutherland1961; Wainwright 1999). Consider the population response to

FIG. 2. Relationship between changes of tuning curves andshifts of perception. Left: tuning curves of 3 neurons withpreferred orientations at �1 � 0 � �2 � �3 � 90 (indicated ingreen, red, and blue, respectively) before adaptation. Thin linesare tuning curves before adaptation, whereas thick lines arethose after adaptation. Green arrows indicate the adaptingorientation �1. Colored dots indicate the firing rates of eachneuron to a test stimulus at �2 after adaptation. Right column:population response curves to the test stimulus. Populationcurves before and after the adaptation are indicated with thingray and thick black lines, respectively. Green arrows indicatethe adapting orientation. Colored vertical lines indicate thefiring rates of the green, red, and blue neurons, respectively,and the colored dots indicate the firing rates of the neurons afteradaptation. Perceived orientations are given by the maxima ofthe population response curves. Pink arrows indicate the direc-tions and magnitudes of shifts of the perceived orientationsafter adaptation. A: adaptation induces only amplitude suppres-sion of the tuning curves. Amplitude suppression is progres-sively smaller because the preferred orientation of the neuron isfurther away from the adapting orientation. Consequently, thepopulation response curve shifts away from the adapting ori-entation, resulting in a repulsive shift of the perceived orienta-tion, as in the tilt aftereffect (TAE). B: adaptation induces onlyrepulsive shifts of preferred orientations of neurons, whichmakes the peak of the population response curve shift towardthe adapting orientation, leading to an attractive shift of theperceived orientation, contrary to the TAE. C: adaptation in-duces both amplitude suppression and repulsive shifts of pre-ferred orientations. Opposing effects of these 2 changes on theperceived orientation leads to a smaller total shift of theperceived orientation than that with suppression alone.

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a test stimulus at �2 � �1 � 0. Before adaptation, the neuronwith label �2 responds maximally to the stimulus because itspreferred orientation matches the stimulus orientation; andneurons with labels equidistant to �2 have the same firing ratesbecause their tuning curves are identical in shape and symmet-ric around the neuron label. Thus the population responsecurve, which describes the firing rates of neurons to thestimulus �2 as a function of their labels, is symmetric around�2; and the perceived orientation of the stimulus is �2. Afteradaptation, the firing rates of neurons with labels in betweenthe adapting and the test orientations are suppressed more thanthose with labels larger than the test orientation. This asym-metric suppression leads to a skewed population responseprofile with the peak position larger than �2, which causes arepulsive shift of the perceived orientation.

Figure 2B shows that the repulsive shift of preferred orien-tation leads to attractive shift of the perceived orientation, ashas been discussed in previous qualitative studies (Gilbert andWiesel 1990; Teich and Qian 2003; Yao and Dan 2001).Consider the firing rate of neuron �2 to the test stimulus at �2.Before adaptation, this neuron has the maximum firing ratecompared to that of other neurons. After adaptation, its pre-ferred orientation shifts to an orientation larger than �2. Thetest stimulus is no longer at the preferred orientation and theneuron’s firing rate drops. The firing rates of all neurons withlabels ��2 also drop because the repulsive shifts make the testorientation farther away from the new preferred orientations.The situation is different for the neurons with labels between 0and �2; the firing rates of some of them actually go up becausethe test orientation is now closer to the new preferred orienta-tions. These asymmetric changes of firing rates lead to askewed population response profile with peak position ��2,which causes an attractive shift of the perceived orientation,opposite to the TAE.

The above discussion demonstrates that the amplitude sup-pression and repulsive shift of preferred orientations haveopposite effects on how the population response and thus theperceived orientation shifts. When both types of change coex-ist, the direction of the perception shift depends on which typepredominates. Figure 2C shows that, if the amplitude suppres-sion predominates, the perceived orientation shifts repulsivelyas in the TAE, but the magnitude of the shift is smaller thanthat with the amplitude suppression alone. On the other hand,if the repulsive shift dominates, the perceived orientation shiftsattractively, opposite to the TAE. Which directions of percep-tion changes are predicted by the physiological data, andwhether the data are consistent with the TAE, can be addressedonly through quantitative analysis of the relative contributionsof the two types of change.

A straightforward way of accessing the consistency betweenthe physiological data and the TAE is as follows: Constructtuning curves of the V1 neurons after adaptation using themeasured amplitude suppressions and preferred orientationshifts, obtain the population response profiles to all stimulusorientations, and calculate the resulting shifts of the perceivedorientations, as shown in Fig. 2C for one stimulus. Thisapproach, although intuitive and simple however, is not thebest way of quantitatively checking the consistency becausethe two types of change lead to opposite effects on theperceived orientation. This situation is analogous to experi-mentally checking whether the relationship A � B � C holds

for three positive variables A, B, C, with A much smaller thanB and C. Data on these quantities are inevitably noisy. Ingeneral, the variance of data is positively correlated to themean, unless the measurements are controlled to high preci-sion; thus, it is reasonable to assume that var (B) and var (C)are much larger than var (A). Note that var (B � C) � var(B) � var (C) and is much larger than var (A). Thus, the meanof B � C can be buried by the variance and is not easilycompared to data on A. A better way is to check the equivalentrelationship B � A � C. Here, var (A � C) � var (A) � var(C). However, because var (A) is much smaller than var (C),the variance of the sum is not too much larger than var (C) andis comparable to var (B). Thus, the mean of A � C is not buriedby the variance, which makes checking the relationship B �A � C far more reliable.

A way of checking the consistency between the physiolog-ical data and the TAE, which is analogous to checking therelationship B � A � C discussed above, is as follows: Use themeasured repulsive shifts of the preferred orientations and themeasured TAE to predict the required amount of amplitudesuppressions of the tuning curves and compare the predictionto the data. This approach takes advantage of the small vari-ance of the TAE measurement (Clifford et al. 2000). In thefollowing, we derive a mathematical relationship that relatesthe shifts of the preferred orientations and the TAE to theamplitude suppressions. This relationship is then used to verifythe consistency between the physiological data and the TAE.

We first define symbols and functions that are useful for ouranalysis. As stated previously, each neuron in the populationhas a label, which is its preferred orientation � in the unadaptedstate. It should be emphasized that � is an invariant label. Afteradaptation, the label of each neuron remains the same as beforeadaptation, even though its preferred orientation may changesubstantially. The rate function F(�, �) is defined as the firingrate of neuron � to a stimulus with orientation �. Note that thetwo Greek letters are mnemonic. Because the orientation of thestimulus is a “physical” quantity, it is denoted by the letter �(“phi”). The label of a neuron is a “psychic” quantity, so it isdenoted by the letter � (“psi”). When considered as a functionof the stimulus orientation � only, F is the tuning curve of theneuron with label �. When considered as a function of theneuron label � only, F is the population response to a stimuluswith orientation �. Therefore the rate function F is a completedescription of both population responses and tuning curves.The rate function can be visualized with a three-dimensionalgraph of firing rate versus neuron label and stimulus orientation(Fig. 3). Two curves on the surface of the rate function areconvenient to define. The first is the perception line �p(�),which marks the perceived orientation of stimulus �. In thewinner-take-all method, �p(�) is computed by maximizing therate function F(�, �) with respect to the neuron label � forfixed stimulus orientation �. The second is the neuron line�n(�), which marks the preferred orientation of neuron �, andis the maximum of the rate function F(�, �) with respect to thestimulus orientation � for fixed neuron label �. Before adap-tation, �n(�) is equal to �, although it may shift after adapta-tion.

Figure 3 illustrates the relationship between the rate function(left) and the neuron and perception lines (right) in varioussituations. Both the population responses (blue) and tuningcurves (red) are shown in the left graphs. The maxima of the





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blue curves in the left graph define the perception line, whichis plotted on the right in blue as a graph of �p(�) versus �. Themaxima of the red curves in the left graph define the neuronline, which is plotted on the right as a graph of �n(�) versus �.

The results of Fig. 3 are the same as those of Fig. 2, butexplained differently through the rate function. This approachfurther leads to a quantitative understanding of the effects ofthe tuning curve changes on orientation perception.

Figure 3A shows the rate function before adaptation, whichlooks like a ridge that is diagonally oriented in the �–� plane(Fig. 3A, left). The perception and neuron lines coincideexactly at the top of the ridge, which means that perceivedorientations are the same as true orientations. When the linesare plotted in two dimensions, they both lie along the diagonal(Fig. 3A, right).

The rest of Fig. 3 illustrates the rate function after adapta-tion. As in Fig. 2, it is helpful to first examine the cases of eachchange happening in isolation (Fig. 3, B and C), then proceedto the case of both changes happening simultaneously (Fig.3D). As before, we assume that the adapting stimulus isoriented at 0 deg.

In Fig. 3B, the preferred orientations shift after adaptation,with no change in tuning-curve amplitudes. In the rate functionon the left, the preferred orientations are the maxima of the redcurves. The neuron line [preferred orientation �n(�) vs. neuronlabel �] is plotted on the right (red). It is shifted away from thediagonal. The shift vanishes at 0 and 90 deg and is maximal atan intermediate orientation. Similar shifts are observed exper-imentally in V1 neurons after adaptation (Dragoi et al. 2000).Because the height of the ridge is constant, the maxima of theblue curves lie along the same line as the maxima of the redcurves. Consequently the perception line (blue) coincideswith the neuron line; both are identically shifted away fromthe diagonal. The shift in the neuron line is repulsive[�n(�) � �], whereas the shift in the perception line isattractive [�p(�) � �].

In Fig. 3C, the tuning-curve amplitudes are suppressed forneurons tuned near the adapting orientation, causing the ridgeof the rate function to be depressed near the origin. There is noshift in preferred orientations, however, so the neuron line liesalong the diagonal, as in the unadapted state of Fig. 3A.Because of the lowered ridge height, the maxima of the bluecurves shift away from the diagonal. As shown on the right, theperception line shifts repulsively [�p(�) � �]. In other words,suppression alone can induce repulsive shifts of perception.

Figure 3D illustrates the effect of combining both suppres-sion and shift, which corresponds to what is observed experi-mentally in V1. The neuron and perception lines both shiftrepulsively. To produce the same perceptual shift as in Fig. 3C,a stronger suppression of the ridge near the origin is necessaryto produce the repulsive shift of the preferred orientation(compare to the rate function of Fig. 3C). The stronger ampli-tude suppression counteracts the weakening effect of the re-pulsive shift of the preferred orientation on the TAE.

In Fig. 3B, the neuron and perception lines coincided ex-actly, and there was no change in the tuning-curve amplitudes.In Fig. 3, C and D, there was a separation between the neuronand perception lines, and the tuning-curve amplitudes weresuppressed near the origin. Figure 4 presents a graphical“proof” that separation between the neuron and perceptionlines is necessarily accompanied by suppression of tuning-curve amplitudes. Consider a path that travels from the redneuron line (point 1) to the blue perception line (point 2) backto the red neuron line (point 3). We will prove that the ratefunction is larger at point 1 than at point 3. The vertical path

FIG. 3. Relationship between changes of tuning curves and shifts of per-ception explained with the rate function. Left: plots of the rate functions. Eachpoint on the surface of a rate function represents the firing rate of a neuron atone stimulus orientation. Here � is the neuron label and � is the stimulusorientation. Only the part of the rate functions for 0 � � � 90 and 0 � � �90 is shown. Adapting orientation is at 0. Red lines are tuning curves and bluelines are the population response curves. Red and blue dots indicate maxima ofthe tuning and population response curves, respectively. Right column: plots ofthe lines of the perceived orientations (blue) and the preferred orientations(red). Lines correspond to the rate function on the left in the following way: thered line represents the locations of the peaks of the tuning curves and the blueline represents the locations of the peaks of the population response curves.Magenta lines indicate that the red and blue lines coincide. Only the parts for0 � � � 90 are shown. A: before adaptation. Tuning curves all have the sameGaussian shape, and the preferred orientations are the same as the neuronlabels. Peaks of tuning curves and population curves coincide. There is no shiftof perception. B: preferred orientations shift repulsively (i.e., they are furtheraway from the adapting orientation compared to their values in the unadaptedstate), whereas the amplitudes remain uniform. Peaks of tuning curves andpopulation curves still coincide. Perceived orientations shift attractively, con-trary to the TAE. C: preferred orientations do not shift, but the amplitudessuppress near the adapting orientation. Peaks of tuning curves and populationcurves no longer coincide. Perceived orientations shift repulsively, as in theTAE. D: preferred orientations shift repulsively and the amplitudes suppressnear the adapting orientation. Suppression is much stronger than in C. This isneeded to produce the same amount of repulsive shift of the perceivedorientations as in C.





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from 1 to 2 travels along the � direction and therefore tracesout the tuning curve of some neuron, which is graphed to theright of the box. Because 1 is on the neuron line, it correspondsto the maximum of the tuning curve, and thus the rate functionat 1 is larger than that at 2. The horizontal path from 2 to 3travels along the � direction, and therefore traces out thepopulation response for some orientation, which is graphedabove the box. Because 2 is on the perception line, it corre-sponds to the maximum of the population response. Thereforethe rate function is larger at 2 than at 3. It follows that the ratefunction at 1 is larger than that at 3. This downhill path can becontinued, alternating between the neuron and perception lines,proving that tuning-curve amplitudes are suppressed near theorigin.

The arguments of Fig. 4 can be made quantitative. Thegeneral idea is to specify the neuron and perception lines andthen derive the amount of amplitude suppression that is re-quired. This calculation can be done by modeling the ratefunction as

F��, �� A�� exp�� n��2

2��2 � (1)

Here the tuning curve of neuron � is Gaussian with amplitudeA(�), width �(�), and preferred orientation �n(�). The func-

tions �n(�) and �(�) are determined by neural data, whereasthe perception line �p(�) is determined by TAE data. From Eq.1 it follows that (see APPENDIX for derivation)

A�� exp��0

�

d��n�� p

�1��

��2 �d�n��

d�

��n�� p

�1��

��

d��

d� �� (2)

Here �p�1(�) is the inverse function of the perception line.

The formula allows the possibility that the widths of tuningcurves vary for different neurons. The adapting orientation is at0. This formula was used to construct Fig. 3. It will also beused in the remainder of this paper, which will quantitativelycompare this predicted amplitude from the model with thetuning-curve amplitudes measured experimentally in V1 neu-rons.

To illustrate Eq. 2 in simple terms, we further simplify itwith two approximations. First, the tuning curves of differentneurons are assumed to have the same width, although thiswidth may change after adaptation. Second, the neuron andperception lines are approximated as piecewise linear. Theapproximate lines coincide with the original neuron and per-ception lines at the origin or 0 deg (the adapting orientation), at90 deg (orthogonal to the adapting orientation), and at loca-tions of maximal shifts, respectively. These approximationsallow us to explicitly express the amplitude of the tuningcurves as a function of five parameters: the width parameter �of the tuning curves; the location � and magnitude � of themaximum repulsive shift of the perceived orientations; and thelocation � and magnitude � of the maximum repulsive shift ofthe preferred orientations (see APPENDIX for details). The am-plitude A(�) of the tuning curve for neuron �near the adaptingorientation at 0 is approximately given by the following ex-pression

A�� exp��

�

�

� �2

2�2 (3)

This formula succinctly summarizes how the amplitude of thetuning curves depends on the shifts of the perception and thepreferred orientation. As the neuron label � goes further awayfrom the adapting orientation at 0, the amplitude grows expo-nentially; the rate of the growth is proportional to the sum ofthe repulsive shifts of the perceived and the preferred orienta-tions and is inversely proportional to the width of the tuningcurves. The rise of the amplitude is more pronounced for large� compared to the case of � � 0 (i.e., no shift of preferredorientation), as illustrated in Fig. 3, D and C.

The analytical result of Eq. 3 was derived from Eq. 2 usingpiecewise-linear approximations for the perception and neuronlines. We can also do the same calculation numerically, usingthe perception line and neuron line determined by the data.Fitting with fourth-order polynomials, we determine the per-ception line �p(�) from the psychophysical data on the TAE(Clifford et al. 2000) (Fig. 5A); fitting with a straight line, wedetermine the neuron line �n(�) from the physiological data(Dragoi et al. 2000) (Fig. 5B). In the physiological experi-ments, adaptation also changes the width of the tuning curves,as evident from the changes of the orientation selectivityindices (OSIs) (Dragoi et al. 2000). We convert the data on the

FIG. 4. Graphic “proof” that repulsive shift of preferred and perceivedorientations is necessarily accompanied by suppression of tuning-curve am-plitudes. In the box, the red line indicates the preferred orientation for theneuron with label � and is termed the neuron line; the blue line indicates theperceived orientation for the stimulus at orientation � and is termed theperception line. Consider a path that travels from the red neuron line (point 1)to the blue perception line (point 2) back to the red neuron line (point 3). Weshow that firing rate is larger at point 1 than that at point 3. Vertical path from1 to 2 travels along the � direction and therefore traces out the tuning curve ofsome neuron, which is graphed to the right of the box. Because 1 is on theneuron line, it corresponds to the maximum of the tuning curve; therefore thefiring rate at 1 is larger than that at 2. Horizontal path from 2 to 3 travels alongthe � direction and therefore traces out the population response for someorientation, which is graphed above the box. Because 2 is on the perceptionline, it corresponds to the maximum of the population response; therefore thefiring rate is larger at 2 than that at 3. It follows that the firing rate at 1 is largerthan that at 3. This downhill path can be continued, alternating between theneuron and perception lines, proving that tuning-curve amplitudes are sup-pressed near the origin. This suppression is large if the distance of the path islong or, equivalently, the repulsive shifts of the preferred orientations and theperceived orientations are large. Narrow tuning curves or population responsecurves have a similar effect.





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OSIs to that on the width parameters assuming a linear rela-tionship between the OSIs and the width parameter (Swindale1998). The relationship is determined by two points: the widthparameter is zero when OSI is 1 and the averaged half-width oftuning curves before adaptation is 30 deg (Watkins and Berk-ley 1974). We fit the converted data on the width parameterswith a straight line (Fig. 5C).

Using the experimentally determined perception and neuronlines and tuning width parameters, we calculate the amplitudeof the tuning curves using Eq. 2. The calculated amplitudecurve is plotted in Fig. 6, left (red curve). We compare thiscurve with physiological data on the amplitude changes of thetuning curves, which are calculated by taking the ratios of theobserved maximum firing rates of neurons after and beforeadaptation (the green circles with error bars in Fig. 6). [These

data are derived from neurons studied by Dragoi et al. (2000),but are presented in this form for the first time. The spontane-ous firing rates are subtracted before calculating the ratios.]The overall scaling of the amplitude curve is not determined inthe theory; we therefore scaled the curve to best match theexperimental data. The comparison is in rough quantitativeagreement (sum of square difference to the data points dividedby variance, or chi-square 2 � 0.27; 2 test of the hypothesisthat there is no difference between the data and the predictionp � 1 � 0.002; degree of freedom for 2 test is 5).

To see whether the observed changes of the tuning curves, inparticular whether the shifts in preferred orientation, are essen-tial for agreement between the theory and the data, we calcu-lated the amplitude curve using the perception line from thepsychophysical data but assuming no shifts of the preferredorientations. The results are plotted in Fig. 6, right (dotted redline). The comparison is clearly not as good as that with thepreferred orientation shifts (2 � 0.83, p � 1 � 0.02).

To summarize, we have compared two models, one thatincludes the repulsive shifts of preferred orientation and onethat does not. Figure 6 shows that the model with preferredorientation shifts explains the data better. This is not the resultof fitting extra parameters to produce better agreement in Fig.6 because the preferred orientation shifts are determined by an

FIG. 6. Consistency of model results with physiological and psychophysi-cal data on adaptation. From the physiological data, we calculate the ratio ofthe amplitudes of the tuning curves after and before adaptation, after subtrac-tion of the spontaneous firing rates (adapted from Dragoi et al. 2000). Greencircles are obtained by averaging these ratios within 15-deg bins, and the errorbars represent SE. Curves are the theoretical calculations of the ratio as afunction of the difference between the neuron label (i.e., preferred orientationbefore adaptation) and the adaptation orientation. Theoretical curves are scaledto minimize the square error relative to the data. Curves are color-coded toindicate the population coding methods used: winner-take-all (red and ma-genta), population vector (black), and maximum-likelihood (blue). Left: solidcurves on the left are calculated with repulsive shifts of both perceived andpreferred orientations, as in Fig. 4D. Except the magenta curve, all curves arecalculated using the results shown herein for shifts of the perceived andpreferred orientations, as well as the changes of the tuning width. Magentacurve is calculated using the piecewise linear approximations to the shifts, anda constant tuning width. Right: dotted curves on the right are calculated withthe shifts of the perceived orientations, but without the repulsive shifts of thepreferred orientations, as in the model of Fig. 4C. Curves on the left are moreconsistent with the physiological data than those on the right. For the amountof the TAE measured in psychophysical experiments, the repulsive shifts ofpreferred orientations are necessary for better explaining the observed amountof relative suppression of the tuning curves.

FIG. 5. Least-squares fits of experimental psychophysical and physiologi-cal data. A: amount of repulsive shift of the perceived orientation is plottedagainst the difference between the testing orientation and the adapting orien-tation. Gray dots are the psychophysical experimental data taken from Cliffordet al. (2000). Black curve is the least-square fit of the data to a polynomial ofthe form �(90 � �)a(1 � b�)(1 � c�). Fitting parameters are a � 0.0061, b ��0.011, c � �0.017. B: amount of repulsive shift of the preferred orientationis plotted against the difference between the neuron label and the adaptingorientation. Gray dots are taken from the physiological experimental data ofDragoi et al. (2000). Black curve is the least-square fit of the data to a line ofthe form a � b�, with parameters a � 13, b � �0.17. C: tuning widthparameter is plotted against the difference between the neuron label and theadapting orientation. Gray dots are adapted from the orientation tuning indexdata of Dragoi et al. (2000). Black line is a least-squares fit of the data to a lineof the form a � b�, with the parameters a � 27, b � �0.34.





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independent fit to data in Fig. 5B. In both the left and rightpanels of Fig. 6, there is only a single free parameter, anoverall scale factor.

These results do not depend on either the exact shapes of theperception and the neuron lines or the variations of the tuningwidth. This is evident from the results that use the formulaassuming piecewise-linear approximations of the perceptionand neuron lines and a constant width for the tuning curves (theformula is given in Eqs. A7 and A8 in the APPENDIX). The casewith the repulsive shifts is plotted in Fig. 6, left (magentacurve, 2 � 0.057, p � 1 � 0.00004) and that without isplotted in Fig. 6, right (dotted magenta line, 2 � 0.65, p �1 � 0.01). The results with the approximations closely resem-ble those without. Parameters for the piecewise-linear approx-imations are as follows: the maximum perceived orientationshift is 4 deg when the testing and adapting angles are 15 degapart (Clifford et al. 2000); the maximum preferred orientationshift is 10 deg when the neuron label and the adapting angle are5 deg apart; and the half-width of the tuning curves is 30 deg(Watkins and Berkley 1974).

We conclude from these comparisons that the observedchanges of the neuron tuning curves after adaptation arequantitatively consistent with the TAE measured psychophys-ically. Moreover, the repulsive shifts of the preferred orienta-tions are especially important for a better quantitative expla-nation of the TAE.

So far we have presented results using the winner-take-allmethod for reading out perceived orientations from the popu-lation responses. Similar results are obtained using two otherreadout methods: the population vector and the maximumlikelihood.

In the population vector method, the perceived orientation ofa stimulus is constructed from the responses of all neurons.Each neuron is assigned a vector, with the length proportionalto the neuron’s firing rate, and the angle with the horizontalaxis equal to twice that of the neuron label (the preferredorientation before adaptation). Summation of the vectors givesa population vector, whose angle is assigned as twice that ofthe perceived orientation. Mathematically, the perceived ori-entation of stimulus � is expressed according to the followingformula

�p�� 1

2arctan ��90

��90

d� sin �2��F��, ��

��90

��90

d� cos �2��F��, �� (4)

This readout method is more complicated than the winner-take-all method, and it is not straightforward to describe closed-form mathematical expressions for the amplitude functionA(�). Nevertheless, we can approximately compute this func-tion by following an optimization procedure.

This is done by parameterization of the amplitude function.For each parameter set, a unique amplitude function is deter-mined. Using the amplitude function, the rate function isconstructed from the tuning curves and is used to calculate theperceived orientations of all stimuli. The shifts of the perceivedorientations relative to the true stimulus orientations are thencompared with the psychophysical data. We search the param-eter space until the best comparison is reached. Some param-

eters can lead to multipeaked population profiles for somestimuli, for which the population vector method breaks down(Pouget et al. 2000); we therefore exclude such parameters inthe search.

Mathematically, the above procedure can be expressed asfollows. We parameterize A(�) with Chebyshev polynomials(Press et al. 1988) up to the fifth order, as below

A�� 1 � k�1

5

ak�cos �k arccos �� 45

45� cos �180k�� (5)

Here ak, k � 1,. . .,5 are the parameters. For each set ofparameters, the perception line �p(�, {ak}) is calculatedthrough Eqs. 5, 1, and 4. We search the parameter space tominimize the following error function

E��ak�� 0

90

d��p��, �ak�� p, OBSERVED��2 � R��ak�� (6)

Here R({ak}) is the regularization factor. This factor is intro-duced to restrict the solution in parameter space so that nomultiple peaks are allowed in the population profiles for anystimulus orientation because the population vector methodbreaks down for multipeaked population profiles. The factor Ris 0 if the population profiles for all stimuli are one-peaked;otherwise, R is assigned a large number so that the error islarge. The exact value of this large number does not affect theresults (in our calculations, the value of the number is 15).Minimization of the error function leads to an amplitudefunction that produces a perception line as close as possible tothe one determined by the psychophysical experiments, withthe constraints that the population activity profiles for allstimuli are one-peaked. We use Powell’s method (Press et al.1988) for minimizing the error function.

In Fig. 6, left, we show the result (black curve) obtainedusing the experimentally determined curves for the shifts of thepreferred and perceived orientations as well as the widthparameters (Fig. 5). The result is quite similar to that of thewinner-take-all method (red curve in Fig. 6), and again com-pares well with the experimental data (2 � 0.089, p � 1 �0.0001). We also calculated the amplitude function withoutrepulsive shifts of the preferred orientations and changes of thetuning width, which is shown in Fig. 6, right (dotted blackcurve). The comparison with the data is less compelling thanthat with the preferred orientation shifts (2 � 0.61, p � 1 �0.01).

The results using the maximum-likelihood method (the bluecurve in Fig. 6, left, with repulsive shifts of the preferredorientations, 2 � 0.14, p � 1 � 0.0004; and the dotted bluecurve in Fig. 6, right, without, 2 � 0.70, p � 1 � 0.02) arealso quite similar to those of the winner-take-all method. In themaximum-likelihood method, the perceived orientation of astimulus � is determined by fitting the population profile withpreset templates, which are the population profiles beforeadaptation. Each perceived orientation has a correspondingtemplate. The template corresponding to a perceived orienta-tion � is a Gaussian function centered at � with the widthparameter �. The fitting procedure minimizes the integral ofthe square error between the template and the populationprofile through scaling the maximum of a template. The labelof the best-matched template is assigned as the perceived





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orientation. To calculate the amplitude function A(�), wefollow a numerical optimization procedure similar to that forthe population vector-average method. A regularization factoris also similarly introduced because the maximum likelihoodmethod is valid only for one-peaked population profiles justlike the vector-average model.

The above results demonstrate that the TAE is consistentwith adaptation-induced changes of the tuning curves of V1neurons. The observed repulsive shifts of the preferred orien-tations after adaptation significantly contribute to such consis-tency. We can show this in another way: from the amplitudefunctions shown in the left graph of Fig. 6 (solid red curve forwinner-take-all, solid black curve for population vector, orsolid blue curve for maximum likelihood), we predict the TAEand compare to the psychophysical data. Specifically, weconstruct the rate function through Eq. 1 with the amplitudefunction A(�), the preferred orientations �n(�), and the widthparameters �(�). From the rate function we obtain the popu-lation response profile for a given stimulus orientation anddetermine the perceived orientation using a population codingmethod (winner-take-all, population vector, or maximum like-lihood). The difference between the perceived and true orien-tations of the stimulus is the predicted TAE. The width param-eters are given in the curve plotted in Fig. 5C. We firstcalculate the TAE with the repulsive shifts of the preferredorientations, given in the curve of Fig. 5B. The results areplotted in Fig. 7, where the solid red, black, and blue curves arefrom winner-take-all, population vector, and maximum-likeli-hood coding methods, respectively. The calculated TAE agreeswith the data (green circles, same as in Fig. 5A) quite well (the

mean square root differences between the data and predictionsare 0.36, 0.35, and 0.44, respectively). We then predict theTAE with no shifts of the preferred orientations. The resultsare plotted in Fig. 7, where the dotted red, black, and blue linesare from winner-take-all, population vector, and maximumlikelihood, respectively. The calculated TAE does not agreewith the data as well (the mean square root differences are 5.1,1.8, and 2.3, respectively). Without the repulsive shifts of thepreferred orientations, the predicted TAE is much larger thanobserved. The repulsive shifts of the preferred orientations thuscontribute significantly to the consistency between adaptation-induced changes of orientation tuning and the TAE.

The amplitude A(�) that predicts a TAE closest to theobserved data is not a linear function, as shown in Fig. 6.However, given relatively large scatter of the physiologicaldata on the amplitude suppression of tuning curves, shownin Fig. 6 as binned average and in Fig. 8A as scatter, suchdeviations from linearity might be hard to distinguish sta-tistically. Thus, it is instructive to calculate the predictedTAE with the amplitude function obtained by fitting straightlines to the suppression data, as shown in Fig. 8A, andcompare to data the predicted TAE with and without repul-sive shifts of preferred orientations. Lines with slopeswithin a range all fit well with the amplitude data, as shownin Fig. 8A, where we shown three fitted lines. The red lineis the result of least-square fit to the data and the black andblue lines are with the slope increased or decreased by 1 SEof the slope estimated in the least-square fit, respectively.The lines are also scaled to best fit the data. The predictedTAE sensitively depends on the amplitude function, asshown in Fig. 8B, where we show TAE predicted from theamplitude lines shown in Fig. 8A (red, black, blue lines areresults from the amplitude lines with corresponding colorsin Fig. 8A; solid lines are from the model with the repulsiveshifts of the preferred orientations and the dotted lines arefrom the model without). However, the overall shape ofpredicted TAE is closer to the actual TAE with the repulsiveshifts. The TAE predicted without the repulsive shifts ingeneral predict more TAE when the difference between thetesting and adapting orientations is large. Because of thesensitivity of the predicted TAE on the amplitude function,we systematically vary the slope of the fitted line within �4to 4 SE of the slope of the least-square-fit line (the red linein Fig. 8A), while also scaling the lines to best fit the data.For each fitted line, we compute, with the maximum-likeli-hood readout method, the predicted TAE, and calculate themean square root difference between the predicted andobserved TAE. The results are shown in Fig. 8C (with therepulsive shifts, sold line; without, dotted line). The differ-ence achieves minimum within �2 SE of the slope bothwhen the repulsive shifts of preferred orientations are in-cluded (solid cyan lines) and when not (dotted cyan lines).However, the minimum value of the difference is smallerwith the repulsive shifts than that without. This can also beseen in Fig. 8D, where we plot the TAE best predicted fromthe model with the repulsive shifts (solid line) and that fromthe model without (dotted line). These results demonstratethat, within the scatter of the amplitude suppression data, themodel with the repulsive shifts produces better prediction ofthe TAE.

FIG. 7. Impact of repulsive shifts of the preferred orientations on accuracyof predicting TAE. Perceived orientation of a stimulus is calculated byobtaining population response profile from the rate function constructed usingthe adaptation-induced changes of tuning curves. Difference between theperceived and true orientations is the TAE. We use the amplitudes of the tuningcurves as given in the left graph (the red, black, or blue curves for winner-take-all, population vector, or maximum-likelihood coding methods, respec-tively). Width parameters are given in the curve of Fig. 6C. Results with therepulsive shifts of the preferred orientations, as given in the curve of Fig. 6B,are plotted as the solid curves. Results with no shifts of the preferredorientations are plotted as the dotted curves. Red, black, or blue indicates thatthe population coding method use is winner-take-all, population vector, ormaximum likelihood, respectively. Green circles are the psychophysical data(same as in Fig. 6A). Although the solid lines agree well with the data, thedotted lines do not. Including the repulsive shifts of preferred orientations isimportant for achieving accurate predictions of TAE with the given amplitudefunctions.





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D I S C U S S I O N

Our results demonstrate that the observed magnitude of theTAE is consistent with the amount of suppression and pre-ferred orientation shift observed in V1 neurons. These twotypes of tuning curve changes have antagonistic effects on theTAE. The functional significance of this antagonism is unclear.One possibility is that relative suppression is an inevitable“design constraint” on neural circuitry and that the shift inpreferred orientations has evolved to counter its effects. It isunlikely that our conclusion depends on the exact details ofhow the rest of the brain reads out the orientation informationfrom the population responses of V1 neurons. This is evidentfrom the agreement among the results using three differentreadout methods.

Past models of the TAE have relied only on the relativesuppression of neural responses (Clifford et al. 2000; Coltheart1971; Dong 1996; Sutherland 1961; Wainwright 1999). Ourcalculations indicate that these models are incomplete becausethey cannot reconcile quantitatively the amount of relativesuppression with the magnitude of the TAE. To produce theobserved magnitude of the perception shift, these models needless suppression than observed experimentally (Fig. 6, right);on the other hand, with the observed amount of suppression,the model predicts too much perceptual shift. Our model

includes not only the suppression but also the repulsive shift ofthe preferred orientations and indicates that these two changesof the tuning curves together lead to a quantitatively consistentmodel of the TAE. Such consistency suggests that the psycho-physical phenomenon of TAE can be explained by the prop-erties of V1 neurons.

In this paper we have focused on the effects of prolongedadaptation (on the order of minutes). Recent physiologicalexperiments have studied the effects of brief adaptation (�1 s)(Chung et al. 2002; Dragoi et al. 2002; Felsen et al. 2002;Muller et al. 1999). These experiments demonstrate that a briefadaptation can also cause the tuning curves to suppress rela-tively and shift repulsively, similar to that during prolongedadaptation. However, the TAE is weak, if any, for briefadaptation (Magnussen and Johnsen 1986). A possibility is thatin this case the repulsive shift cancels the shift of perceptionthat can arise from the relative suppression, resulting in a weakTAE. This hypothesis can be quantitatively tested in the sameway as in this paper by comparing the observed amount ofsuppression to that calculated with the observed shifts of thepreferred orientations alone.

Previously, while simulating the tilt illusion, a psycho-physical phenomenon related to the TAE, Gilbert and Wiesel(1990) observed the opposing effects of amplitude suppres-

FIG. 8. Prediction errors of TAE with the adaptation-induced amplitude changes of tuning curves fitted with straight lines. A: scatterplot of theadaptation-induced change of amplitude A(�) of neuron with label � (gray dots). These data are presented in Fig. 5 (green circles) by averaging over intervalsof 15 deg. Three straight lines are shown. Red line is the least-square fit to the data (form a � b�, where a � 0.63, b � 0.0066); black is with the slope increasedby a SE of the slope (0.0015) calculated in the least-square fit, and the overall scale adjusted to best fit the data; blue is the same as the black, but with the slopedecreased by a SE. B: predicted TAE with (solid lines) and without (dotted lines) the repulsive shifts, compared with the actual TAE (green dots). Red, black,and blue lines are calculated with amplitude lines of corresponding colors shown in A. Overall angular dependency of the predicted TAE compares better to theactual TAE with the repulsive shifts than without. C: comparison of the prediction errors of TAE with (solid cyan line) and without (dotted cyan line) the repulsiveshifts of the preferred orientations. Calculation of the TAE is the same as in Fig. 7, except that A(�) is fitted with straight lines. Slope of the fitting line issystematically varied by �4 to 4 SE. Overall scale of the line is adjusted to best fit the amplitude data. For each line, TAE is calculated with and without repulsiveshifts of the preferred orientations, and the prediction error is computed as the mean square root difference from the actual TAE data (green circles).Maximum-likelihood method is used as the readout method of perceived orientation. Minimum prediction error achieved is smaller with repulsive shifts thanwithout. D: TAE best predicted in C. Result from the model with repulsive shifts of the preferred orientations (solid cyan line) fits better to the data (green dots)that that from the model without (dotted cyan line).





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sion and repulsive shifts of preferred orientations on orien-tation perception. Our mathematical analysis goes beyondthis qualitative observation and unveils a quantitative rela-tionship between the suppression and the shifts of percep-tion and preferred orientations. Such relationships enable usto check the consistency between psychophysical and phys-iological data. The relationship is also useful as a cautionagainst an oversimplified conclusion that the directions ofshifts in the tuning curves and the perceived orientations arealways opposite (Teich and Qian 2003; Yao and Dan 2001).As our analysis shows, the relative directions depend on thechange of the amplitudes of the tuning curves. Inferringperception from tuning curves must be done through acareful analysis of the firing rate function constructed fromthe tuning curves.

We briefly note some general constraints on the source ofthe suppression. Suppression in V1 neurons could arise fromchanges within the cortex itself or from changes in theinputs to the cortex. The latter type of change, although nodoubt present, does not seem to be important for the psy-chological and physiological experiments considered in thispaper. Adaptation is induced with a drifting grating, whichtends to activate all LGN neurons equally over time becausethey are at most weakly tuned to orientation (Hubel andWiesel 1961). Therefore, any adaptation of thalamocorticalinput is expected to be unspecific for orientation. Thus, thelocus of orientation-specific adaptation is likely to be thecortex (Carandini and Heeger 1994; Movshon and Lennie1979). One plausible scenario involves adaptation of theexcitatory connections between neurons tuned to the adapt-ing orientation, which would cause the responses of theseneurons to be suppressed (Dong 1996; Felsen et al. 2002;Teich and Qian 2003). Quantitative support in favor of thisidea comes from independent studies of the effects ofrecurrent excitation. Recurrent excitation is estimated toamplify cortical responses by a factor of about two or three(Ferster et al. 1996). Therefore, weakening of recurrentexcitation by adaptation would be expected to reduce theamplitudes of tuning curves by up to this factor, which is inrough quantitative accord with the tuning-curve suppressiondepicted in Fig. 6. Another possible source of adaptationwithin the cortex is reduction in spiking of neurons attrib-uted to activation of long-lasting hyperpolarizing currentsafter prolonged spiking activity (Carandini and Ferster1997; Sanchez-Vives et al. 2000).

We have shown that the TAE is consistent with thephysiological data obtained from anesthetized cat V1 neu-rons. Two caveats should be noted. First, the psychophysicsof the TAE is based on human experiments. However, thefunction and organization of V1 is quite similar acrossmammalian species and V1 neurons likely behave similarlyunder adaptation. Indeed, several recent studies have dem-onstrated neural correlates of high-level perception in hu-man V1 (Lee et al. 2005; Ress and Heeger 2003). It mightalso be possible to test the TAE and measure V1 neuronresponses in the same animal. Quantitative consistency ofthe psychophysical and physiological data in such experi-ments should provide a stronger ground for concluding thatthe TAE arises from the properties of V1 neurons. Second,our model relies on a simple assumption: the unadaptedtuning curves of neurons are the same and adaptation-

induced changes of the tuning curves depends only on thedistance between the labels of the neurons and the adaptingorientation. This assumption, which is also the basis of mostof the previous models of the TAE (Clifford et al. 2000;Coltheart 1971; Sutherland 1961; Wainwright 1999), en-ables us to derive in compact form the quantitative relation-ship between the changes of the tuning curves and the TAE.In reality, the tuning curves have a wide range of shapes andadaptation-induced changes of these curves also depend onthe locations of neurons relative to the orientation map(Dragoi et al. 2001). Our model does not account for theobserved diversity of neuron properties. It remains to beseen how inclusion of this diversity, which inevitably re-quires large-scale simulations of the visual cortex, mightmodify our results.

Besides the TAE, there are other adaptation-induced af-tereffects such as the motion aftereffect (Huk et al. 2001)and the spatial frequency aftereffect (Humanski and Wilson1993). Population coding models similar to the one in thispaper can also be useful for quantitatively checking theconsistency between psychophysical and physiological datain these aftereffects. A recent study of motion adaptation inMT neurons in anesthetized monkeys demonstrates an at-tractive shift in the direction tuning of these neurons, towardthe adapting direction (Kohn and Movshon 2003, 2004).Whether such an attractive shift in direction tuning gener-alizes to other kinds of adaptation, such as orientationadaptation, in visual cortical areas remains to be examined.However, these findings raise the issue of the perceptuallocus of adaptation-induced changes, in particular whetherspecific visual areas are privileged sites for specific perceptsand whether there are different mechanisms and conse-quences of pattern adaptation in different cortical areas. Amore complete description of neuronal responses and theconsequences of pattern adaptation in different areas ofvisual cortex will be required to resolve these issues.

A P P E N D I X

In the following we describe the procedures for calculating A(�)with the winner-take-all method. For convenience, we let the adaptingorientation be at 0, and discuss only the case of � � 0 (the case for� � 0 is similar). The overall scaling of A(�) is not determined fromthe calculations. Here we simply set A(0) � 1.

In the winner-take-all method, the label of the neuron that fires themost in the population equals the perceived orientation. The popula-tion response curve to a stimulus � is given by F(�, �) with � heldconstant. The peak location of this curve is given by setting to zero thepartial derivative of the rate function with respect to the neuron label,that is

0 �F��, ��

��

dA��

d�exp��

�� n��2

2��2 �� A�� exp�� n��2

2��2 �� n��

��2

d�n��

d��

�� n��2

��3

d��

d�� (A1)

Here we used Eq. 1 for the rate function F(�, �). Rearranging termsin above equation, we find

dA��

d�� A��

�n��

��2 �d�n��

d��

�n��

��

d��

d� (A2)





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Solving the above equation for �, we obtain the perceived orien-tation of stimulus �. In other words, the solution of the aboveequation is given by � � �p(�). An equivalent expression of thesolution is � � �p

�1(�), where �p�1 is the inverse function of the

perception line �p(�). Therefore, the above equation can also be

written as

dA��

d�� A��

�n�� p�1��

��2 �d�n��

d��

�n�� p�1��

��

d��

d� (A3)

This equation gives the relationship among the amplitude functionA(�), the perception line �p(�), the neuron line �n(�), and the widthparameter �(�). Notice that the changes of the amplitudes of thetuning curves, depicted by the derivative on the left-hand side of the

above equation, cannot be arbitrary. Noting that

1

A��

dA��

d��

d ln A��

d�

we solve the above differential equation to find

A�� exp��0

�

d��n�� p

�1��

��2 �d�n��

d�

��n�� p

�1��

��

d��

d� �� (A4)

Numerically integrating the above equation gives the amplitude func-tion A(�) with given neuron and perception lines, as well as the tuningwidth parameters.

For the adapted state, we can explicitly evaluate Eq. A4 withpiecewise-linear approximations to the neuron and the perceptionlines and assuming that the tuning width parameter is a constant �.The neuron line is approximated with two straight lines determinedby three pairs of neuron label and the preferred orientation: (0, 0),( , � �), (90, 90), where is the label of the neuron that showsthe maximum preferred orientation shift �. This gives the following

expression for the approximated neuron line

�n�� k1� for 0 � � � 90 � k2�90 � �� for � � � 90

(A5)

Here

k1 � � �

and k2 �

� � � 90

� 90

are the slopes of the two lines. Similarly, the perception line isapproximated with two straight lines determined by three pairs of thestimulus orientation and the perceived orientation: (0, 0), (�, � � �),(90, 90), where � is the orientation of the stimulus that shows themaximum perception shift �. The inverse function of the approxi-

mated perception lines is given by the following equation

�p�1�� k3� for 0 � � � � � �

90 � k4�90 � �� for � � � � � � 90(A6)

Here

k3 ��

� � �and k4 �

� � 90

� � � � 90

are the slopes of the two lines of the inverse function. Plugging in Eqs.A5 and A6 into Eq. A4 yields the following expression for theamplitude modulation:

Case 1: � � �.

In this case, we have

A�� exp�a�2

2�2 for 0 � � �

exp�b�2 � c� � d

2�2 for � � � � �

exp�f�2 � g� � h

2�2 for � � � � � 90 (A7)

Here the constants are given by a � k1 (k1 � k3), b � k2 (k2 � k3),c � 180k2 (1 � k2), d � 902 (1 � k2)2 � k3 (k2 � k1) 2, f � k2(k2 �k4), g � 180k2 (k4 � k2), and h � k3 (k2 � k1) 2 � k2 (k4 � k3)(� ��)2 � 902 (1 � k2) 2 � 180k2 (1 � k4)(� � �).

Case 2: � � � � .

In this case, we have

A�� exp�a�2

2�2 for 0 � � � � � �

exp�u�2 � v� � w

2�2 for � � � � �

exp�f�2 � g� � s

2�2 for � � 90 (A8)

Here the constants are given by u � k1 (k1 � k4), v � �180k1 (1 �k4), w � k1 (k4 � k3)(� � �)2 � 180k1 (1 � k4)(� � �), and s � k1

(k4 � k3)(� � �)2 � k4 (k2 � k1) 2 � 902 (1 � k2)2 � 180(1 �k4)(k1 (� � �) � (k2 � k1) ).

As can be seen from Eqs. A7 and A8, piecewise-linear approxima-tions of the neuron and the perception lines result in a piecewise-Gaussian surface for the rate function. Away from the adaptingorientation at 0, the amplitude function increases exponentially withthe square of the distance of the neuron label from the adaptingorientation. The rate factor of this exponential can be expressed asfollows when the shifts of the perceptions and preferred orientationsare small compared to � and , respectively

a

2�2 �1

2�2 ��

�

�

� (A9)

The above equation shows that larger repulsive shifts of perceptionand repulsive shifts of the preferred orientation add up to cause afaster rise of the amplitudes of the tuning curves away from theadapting orientation.

G R A N T S

D. Z. Jin was supported by the Huck Institute of Life Sciences at ThePennsylvania State University, H. S. Seung was supported by Howard HughesMedical Institute, V. Dragoi was supported by the McDonnell–Pew Founda-tion, and M. Sur received support from the National Institutes of Health.

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